The Projective Line

Abstract

In this chapter K is an algebraically closed field, complete with respect to a nonarchimedean valuation. The projective line P over K is, as usual, K ⋃ {∞} and z denotes the variable of P. The goal of this chapter is to develop the function theory on suitable subsets of P with a minimum of mathematical tools. This elementary level is already sufficient for the exposition of D. Harbater’s theorem, [125], concerning the Galois groups of extensions of the field Q _p (z). Moreover, the rigid analytic part of M. Raynaud’s proof of Abhyankar’s conjecture for the affine line in positive characteristic, [213], can be understood without further knowledge of rigid analytic spaces. We will present these proofs in Chapter 9.